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Centers of Triangles or Points of Concurrency. Geometry October 28, 2013. OBJECTIVE. You will learn how to construct perpendicular bisectors, angle bisectors, medians and altitudes of triangles constructed. Today’s Agenda. Triangle Segments Median Altitude Perpendicular Bisector

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Presentation Transcript
objective
OBJECTIVE

You will learn how to construct perpendicular bisectors, angle bisectors, medians and altitudes of triangles constructed

today s agenda
Today’s Agenda
  • Triangle Segments
    • Median
    • Altitude
    • Perpendicular Bisector
    • Angle Bisector
  • Triangle Centers
    • Centroid
    • Orthocenter
    • Circumcenter
    • Incenter
vocabulary
VOCABULARY
  • A median of a triangle is a segment that connects a vertex to the midpoint of the opposite side.
  • The altitude of a triangle is the perpendicular distance from one of its bases to the opposite vertex. In other words, the altitude is a segment that is perpendicular to one side and reaches the point across from that side.
  • A perpendicular bisector of a line segment isa) perpendicular to it, and b) bisects it.
  • An angle bisector of a triangle is a segment that divides one of its angles into two congruent pieces. The segment connects to the opposite side
slide5

Medians

Median

vertex to midpoint

slide6

Example 1

M

D

P

C

What is NC if NP = 18?

MC bisects NP…so 18/2

9

N

If DP = 7.5, find MP.

15

7.5 + 7.5 =

slide7

How many medians does a triangle have?

Three – one from each vertex

slide8

The medians of a triangle are concurrent.

The intersection of the medians is called the CENTRIOD.

They meet in a single point.

slide9

Theorem

The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint.

2x

x

in abc an bp and cm are medians

Example 2

In ABC, AN, BP, and CM are medians.

If EM = 3, find EC.

C

EC = 2(3)

N

P

E

EC = 6

B

M

A

in abc an bp and cm are medians1

Example 3

In ABC, AN, BP, and CM are medians.

If EN = 12, find AN.

C

AE = 2(12)=24

AN = AE + EN

N

P

E

AN = 24 + 12

B

AN = 36

M

A

in abc an bp and cm are medians2

C

N

P

E

B

M

A

Example 4

In ABC, AN, BP, and CM are medians.

If EM = 3x + 4 and CE = 8x, what is x?

x = 4

in abc an bp and cm are medians3

C

N

P

E

B

M

A

Example 5

In ABC, AN, BP, and CM are medians.

If CM = 24 what is CE?

CE = 2/3CM

CE = 2/3(24)

CE = 16

slide14

Angle Bisector

Angle Bisector

vertex to side cutting angle in half

slide15

Example 1

W

X

1

2

Z

Y

slide16

Example 2

F

I

G

5(x – 1) = 4x + 1

5x – 5 = 4x + 1

x = 6

H

slide17

How many angle bisectors does a triangle have?

three

The angle bisectors of a triangle are ____________.

concurrent

The intersection of the angle bisectors is called the ________.

Incenter

the incenter is the same distance from the sides of the triangle
The incenter is the same distance from the sides of the triangle.

Point P is called the __________.

Incenter

example 4

A

8

D

F

L

C

B

E

Example 4

LF, DL, EL

The angle bisectors of triangle ABC meet at point L.

  • What segments are congruent?
  • Find AL and FL.

Triangle ADL is a right triangle, so use Pythagorean thm

AL2 = 82 + 62

AL2 = 100

AL = 10

FL = 6

6

slide20

Altitude

Altitude

vertex to opposite side and perpendicular

slide21

Tell whether each red segment is an altitude of the triangle.

The altitude is the “true height” of the triangle.

YES

NO

YES

slide22

How many altitudes does a triangle have?

Three

The altitudes of a triangle are concurrent.

The intersection of the altitudes is called the ORTHOCENTER.

slide23

Perpendicular Bisector

Perpendicular Bisector

midpoint and perpendicular

(don't care about no vertex)

slide25

Example 2: Find x

3x + 4

5x - 10

x = 7

slide26

How many perpendicular bisectors does a triangle have?

Three

The perpendicular bisectors of a triangle are concurrent.

The intersection of the perpendicular bisectors is called the CIRCUMCENTER.

find da

Example 3: The perpendicular bisectors of triangle ABC meet at point P.

Find DA.

DA = 6

BA = 12

  • Find BA.
  • Find PC.

PC = 10

  • Use the Pythagorean Theorem to find DP.

B

6

DP2 + 62 = 102

DP2 + 36 = 100

DP2 = 64

DP = 8

10

D

P

A

C

slide29

Tell if the red segment is an altitude, perpendicular bisector, both, or neither?

NEITHER

ALTITUDE

PER. BISECTOR

BOTH

slide30

Sum It Up

bisector

bisector

which is…

Figure

concurrent at..

circumcenter

equidistant from vertices

incenter

equidistant from sides

median

centroid

2/3 distance from vertices to midpoint

altitude

orthocenter

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